In the related art, an automatically-closing door hinge configured to rotate an opened door in a closing direction automatically using a restoring force of a coil spring and provided with a shock absorbing function for absorbing a shock applied when the door is closed using a hydraulic cylinder is known and, in recent years, an automatically-closing door hinge configured to absorb a shock with an air damper instead of the hydraulic cylinder is proposed. For example, disclosed in Patent Document 1 (JP-A-2002-303072) and Patent Document 2 (JP-A-2005-113682) are an automatically-closing door hinge on the basis of an air damper including a piston stored and arranged in a cylinder provided on one of a pair of vanes, and an operating rod fixed to an upper portion of the other vane and arranged within the cylinder, wherein spheres disposed at a predetermined position of the piston so as to roll over and projecting from an inner periphery thereof are engaged with a cam groove being formed on an outer periphery of a lower portion of the operating rod and having an inclined portion, the piston is moved forward and backward corresponding to the movement of the spheres with respect to the inclined portion of the cam groove, and a shock is absorbed by an air cushioning in the cylinder on the basis of a returning action of the piston when the door is closed.